Lecture
An important stage in constructing a mathematical model of the object under study is the formalization of information describing the object's attributes, which may be heterogeneous in terms of the form of their representation. For example, expert knowledge most often cannot be expressed in an exact numerical form suitable for further mathematical analysis, but can only be represented as qualitative characteristics or logical conclusions. Nevertheless, thanks to the use of reasoning set out at a qualitative level, many factors of uncertainty arising in the process of making managerial decisions in an organization can be formalized and correctly taken into account.
For an adequate assessment of heterogeneous object attributes (continuous, discrete, qualitative or quantitative), the mathematical theory of measurement has developed various types of measurement scales.
Definition
In measurement theory, scale is understood as a homomorphism f of the empirical relational system M into the symbolic system H. Thus, a scale is a rule defined by the triple <M, H, f>. Models, images of the real system in the symbolism of the formal system, are called «scale values». The procedure of comparing objects by selected attributes is called measurement, or measurement on a given scale .
Another common definition holds that a scale — is a set of numbers or symbols by means of which a certain characteristic, property of an object or phenomenon, can be measured.
The following types of scales are distinguished .
1. Nominal scale (scale of names). It allows establishing a one-to-one correspondence between objects. The nominal scale is based on the equivalence relation and is used to assign the object under study to a certain class. Arithmetic operations are not defined for it. It is not possible to identify relations such as «greater», «better», «more preferable» or «less preferable», etc.
Examples: names of countries and regions, their telephone codes and postal codes, addresses of construction sites, names of investment projects or units of a mutual investment fund. Naturally, it is impossible to calculate the difference between the fund names «Summer» and «Youth», or to order them, for example, by degree of attractiveness to the investor.
2. Ordinal or rank scale. It is quite often used in applied problems, since it allows not only dividing objects into classes, but also ordering them according to the degree of increase or decrease of a certain property of the object. For the ordinal scale, any monotonic transformation is admissible; arithmetic operations make no sense within it.
Examples: various kinds of ratings (of world universities, the reliability of financial corporations or countries, etc.), knowledge-level grading scales in the education system, product quality assessment scales expressed in natural language. For example, with regard to the characteristics of investment and construction projects, when using an ordinal scale, prospective construction sites can be ordered according to their expected construction complexity: «low», «medium», «above medium», «high», «very high». In this case, it is impossible to indicate the degree of superiority of one object over another.
3. Interval scale. This scale allows not only classifying and ordering objects, but also quantitatively assessing the difference between their characteristics. To carry out such comparisons, the concepts of «scale factor» — a and «reference point» — b. are introduced. Admissible within this scale is the linear transformation: f(x) = ax + b.
Examples: time-measurement scales using different units (second, minute, hour, etc.) and non-coincident reference points, temperature scales (Celsius and Fahrenheit: T = 9/5°C + 32).
4. Ratio scale. It is a special case of the interval scale, for which a certain scale factor is introduced (a > 0), and the reference point equals 0. This scale allows answering the question: how many times one object exceeds another with respect to the selected characteristic.
Examples: scales used to measure distance (in inches, meters, miles), mass (in grams, pounds), temperature (Celsius and Réaumur scales: °Re = °C-4/5), cost (in various currencies).
5. Absolute scale. It is defined by a one-to-one correspondence: f(x) = x and represents a sequence of natural numbers. This scale is used to measure the quantity of objects.
Please note!
The narrower the set of admissible transformations, the more perfect the scale is considered to be, in terms of the degree of informativeness and detail regarding the measured property of the object. Indicators having a scale no less perfect than the interval scale are called quantitative and allow algebraic operations.
If the analyst is interested in selecting the object with the largest value of a certain characteristic, and it does not matter what that value actually equals, then an ordinal scale can be used. If there is a need to select an object whose characteristic value is closest to some given quantity, this characteristic must be treated as quantitative, measured on an interval scale.
Examples of the use of scales
Using the terminology and problem area of investment and construction projects (ICP), the following examples of the use of scales of various types can be given: a nominal scale is used to assign names to ICPs, an ordinal scale — for example, to identify the construction site most attractive to buyers, a ratio scale — when assessing the investment required to implement the project, measured in Russian rubles or another currency depending on the strategy for attracting financial resources and the situation on the real estate market.
Characteristics of scales of various types
|
Scale type |
Possible transformations |
Nature of information |
Nature of the scale |
|
Nominal |
(x = y) = f(x) =f(y) |
Qualitative |
Discrete |
|
Ordinal |
(x<y) = [f(x) |
||
|
Interval |
f(x) = ax + b, where a > 0 |
Quantitative |
Discrete/continuous |
|
Ratio |
f(x) = ax, where a > 0 |
Discrete/continuous |
|
|
Absolute |
f(x) = X |
Discrete |
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